Properties

Label 507.2.j.i
Level $507$
Weight $2$
Character orbit 507.j
Analytic conductor $4.048$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{11} + \beta_{8} - \beta_1) q^{2} + \beta_{7} q^{3} + ( - \beta_{9} - 3 \beta_{7} - \beta_{4} + 3) q^{4} + (\beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 3 \beta_{6} + \beta_{2}) q^{5} + ( - 2 \beta_{2} - \beta_1) q^{6} + ( - \beta_{6} + \beta_{2}) q^{7} + (3 \beta_{11} - 2 \beta_{10} + 3 \beta_{8} + 2 \beta_{6} + 3 \beta_{2}) q^{8} + (\beta_{7} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{11} + \beta_{8} - \beta_1) q^{2} + \beta_{7} q^{3} + ( - \beta_{9} - 3 \beta_{7} - \beta_{4} + 3) q^{4} + (\beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 3 \beta_{6} + \beta_{2}) q^{5} + ( - 2 \beta_{2} - \beta_1) q^{6} + ( - \beta_{6} + \beta_{2}) q^{7} + (3 \beta_{11} - 2 \beta_{10} + 3 \beta_{8} + 2 \beta_{6} + 3 \beta_{2}) q^{8} + (\beta_{7} - 1) q^{9} + (4 \beta_{9} - \beta_{7} - 6 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 4) q^{10} + (2 \beta_{11} + 2 \beta_{10} - \beta_{8} + \beta_1) q^{11} + ( - \beta_{3} + 4) q^{12} + ( - 2 \beta_{3} - 1) q^{14} + (\beta_{11} + 3 \beta_{10} + 2 \beta_{8} - 2 \beta_1) q^{15} + ( - 5 \beta_{9} - \beta_{5} - 6 \beta_{4} + 5 \beta_{3} - 5) q^{16} + (\beta_{9} - \beta_{7} + 3 \beta_{4} + 1) q^{17} + ( - 2 \beta_{11} - \beta_{8} - 2 \beta_{2}) q^{18} + ( - 3 \beta_{6} + \beta_{2} + \beta_1) q^{19} + ( - 8 \beta_{6} + 2 \beta_{2} + 7 \beta_1) q^{20} + (\beta_{11} + \beta_{10} - \beta_{6} + \beta_{2}) q^{21} + (5 \beta_{9} - 7 \beta_{7} + 7 \beta_{4} + 7) q^{22} + ( - \beta_{9} - \beta_{7} + 5 \beta_{5} + 4 \beta_{4} + \beta_{3} - 1) q^{23} + (3 \beta_{11} - 2 \beta_{10} + 3 \beta_{8} - 3 \beta_1) q^{24} + (3 \beta_{5} - 2 \beta_{3} - 4) q^{25} - q^{27} + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{8} - \beta_1) q^{28} + ( - 2 \beta_{9} + \beta_{7} + 3 \beta_{5} + \beta_{4} + 2 \beta_{3} - 2) q^{29} + (4 \beta_{9} - \beta_{7} - 2 \beta_{4} + 1) q^{30} + ( - 2 \beta_{11} - 5 \beta_{10} + 3 \beta_{8} + 5 \beta_{6} - 2 \beta_{2}) q^{31} + (5 \beta_{6} + 7 \beta_1) q^{32} + (2 \beta_{6} - 2 \beta_{2} + \beta_1) q^{33} + ( - \beta_{11} - 7 \beta_{8} - \beta_{2}) q^{34} + ( - 4 \beta_{9} + 3 \beta_{7} - \beta_{4} - 3) q^{35} + (\beta_{9} + 3 \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{36} + (\beta_{11} + 8 \beta_{10} + \beta_{8} - \beta_1) q^{37} + ( - 4 \beta_{5} - 5 \beta_{3} + 3) q^{38} + ( - 8 \beta_{5} - \beta_{3} + 4) q^{40} + ( - 2 \beta_{11} - 4 \beta_{10} + \beta_{8} - \beta_1) q^{41} + (2 \beta_{9} - 3 \beta_{7} + 2 \beta_{4} - 2 \beta_{3} + 2) q^{42} + (2 \beta_{9} + 3 \beta_{7} + 4 \beta_{4} - 3) q^{43} + (3 \beta_{11} + 4 \beta_{10} - 7 \beta_{8} - 4 \beta_{6} + 3 \beta_{2}) q^{44} + (3 \beta_{6} - \beta_{2} - 2 \beta_1) q^{45} + (7 \beta_{6} - 2 \beta_{2} - 12 \beta_1) q^{46} + ( - \beta_{11} - 3 \beta_{10} - \beta_{8} + 3 \beta_{6} - \beta_{2}) q^{47} + ( - 5 \beta_{9} - 6 \beta_{4}) q^{48} + ( - 2 \beta_{9} - 4 \beta_{7} + \beta_{5} - \beta_{4} + 2 \beta_{3} - 2) q^{49} + ( - 7 \beta_{11} - \beta_{10} + 7 \beta_{8} - 7 \beta_1) q^{50} + ( - 2 \beta_{5} + \beta_{3}) q^{51} + (3 \beta_{5} + 4 \beta_{3} - 8) q^{53} + ( - 2 \beta_{11} - \beta_{8} + \beta_1) q^{54} + (10 \beta_{9} - 5 \beta_{7} - 8 \beta_{5} + 2 \beta_{4} - 10 \beta_{3} + 10) q^{55} + ( - \beta_{9} + \beta_{7} - 3 \beta_{4} - 1) q^{56} + (\beta_{11} + 3 \beta_{10} + \beta_{8} - 3 \beta_{6} + \beta_{2}) q^{57} + (7 \beta_{6} - 3 \beta_{2} - 6 \beta_1) q^{58} + (4 \beta_{6} - 2 \beta_{2} - 4 \beta_1) q^{59} + (2 \beta_{11} + 8 \beta_{10} + 7 \beta_{8} - 8 \beta_{6} + 2 \beta_{2}) q^{60} + ( - 5 \beta_{9} - 3 \beta_{7} + \beta_{4} + 3) q^{61} + ( - 13 \beta_{9} + 9 \beta_{7} - 3 \beta_{5} - 16 \beta_{4} + 13 \beta_{3} + \cdots - 13) q^{62}+ \cdots + ( - 2 \beta_{11} - 2 \beta_{10} + \beta_{8} + 2 \beta_{6} - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 22 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 22 q^{4} - 6 q^{9} - 2 q^{10} + 44 q^{12} - 20 q^{14} - 22 q^{16} - 2 q^{17} + 18 q^{22} - 44 q^{25} - 12 q^{27} + 4 q^{29} + 2 q^{30} - 8 q^{35} + 22 q^{36} + 12 q^{40} - 10 q^{42} - 30 q^{43} + 22 q^{48} - 30 q^{49} - 4 q^{51} - 68 q^{53} - 6 q^{55} + 2 q^{56} + 26 q^{61} - 4 q^{62} + 36 q^{66} + 26 q^{68} - 30 q^{74} - 22 q^{75} - 60 q^{77} + 12 q^{79} - 6 q^{81} - 6 q^{82} - 4 q^{87} + 26 q^{88} + 4 q^{90} + 28 q^{92} - 42 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 19x^{8} - 28x^{6} + 31x^{4} - 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -25\nu^{11} + 95\nu^{9} - 361\nu^{7} + 155\nu^{5} - 30\nu^{3} - 1563\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25\nu^{10} - 95\nu^{8} + 361\nu^{6} - 155\nu^{4} + 30\nu^{2} + 1004 ) / 559 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{10} - 20\nu^{8} + 76\nu^{6} - 139\nu^{4} + 124\nu^{2} - 24 ) / 43 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 45\nu^{10} - 171\nu^{8} + 538\nu^{6} - 279\nu^{4} + 54\nu^{2} + 242 ) / 559 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 70\nu^{11} - 266\nu^{9} + 899\nu^{7} - 434\nu^{5} + 84\nu^{3} + 1246\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 114\nu^{10} - 545\nu^{8} + 2071\nu^{6} - 2831\nu^{4} + 3379\nu^{2} - 95 ) / 559 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 114\nu^{11} - 545\nu^{9} + 2071\nu^{7} - 2831\nu^{5} + 3379\nu^{3} - 95\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -128\nu^{10} + 710\nu^{8} - 2698\nu^{6} + 4483\nu^{4} - 4402\nu^{2} + 852 ) / 559 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -242\nu^{11} + 1255\nu^{9} - 4769\nu^{7} + 7314\nu^{5} - 7781\nu^{3} + 1506\nu ) / 559 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -317\nu^{11} + 1540\nu^{9} - 5852\nu^{7} + 8338\nu^{5} - 9548\nu^{3} + 1848\nu ) / 559 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 3\beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} + 2\beta_{7} + 4\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{11} - \beta_{10} + 9\beta_{8} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{5} + 9\beta_{3} - 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{6} - 14\beta_{2} - 28\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -28\beta_{9} - 14\beta_{7} - 19\beta_{5} - 47\beta_{4} + 28\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -47\beta_{11} + 19\beta_{10} - 89\beta_{8} - 19\beta_{6} - 47\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -89\beta_{9} - 42\beta_{7} - 155\beta_{4} + 42 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -155\beta_{11} + 66\beta_{10} - 286\beta_{8} + 286\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(1 - \beta_{7}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
1.56052 0.900969i
−1.07992 + 0.623490i
−0.385418 + 0.222521i
0.385418 0.222521i
1.07992 0.623490i
−1.56052 + 0.900969i
1.56052 + 0.900969i
−1.07992 0.623490i
−0.385418 0.222521i
0.385418 + 0.222521i
1.07992 + 0.623490i
−1.56052 0.900969i
−2.33136 1.34601i 0.500000 0.866025i 2.62349 + 4.54402i 1.04892i −2.33136 + 1.34601i −0.480608 + 0.277479i 8.74094i −0.500000 0.866025i −1.41185 + 2.44540i
316.2 −2.04113 1.17845i 0.500000 0.866025i 1.77748 + 3.07868i 3.69202i −2.04113 + 1.17845i 0.694498 0.400969i 3.66487i −0.500000 0.866025i 4.35086 7.53590i
316.3 −1.77441 1.02446i 0.500000 0.866025i 1.09903 + 1.90358i 3.35690i −1.77441 + 1.02446i 1.94594 1.12349i 0.405813i −0.500000 0.866025i −3.43900 + 5.95652i
316.4 1.77441 + 1.02446i 0.500000 0.866025i 1.09903 + 1.90358i 3.35690i 1.77441 1.02446i −1.94594 + 1.12349i 0.405813i −0.500000 0.866025i −3.43900 + 5.95652i
316.5 2.04113 + 1.17845i 0.500000 0.866025i 1.77748 + 3.07868i 3.69202i 2.04113 1.17845i −0.694498 + 0.400969i 3.66487i −0.500000 0.866025i 4.35086 7.53590i
316.6 2.33136 + 1.34601i 0.500000 0.866025i 2.62349 + 4.54402i 1.04892i 2.33136 1.34601i 0.480608 0.277479i 8.74094i −0.500000 0.866025i −1.41185 + 2.44540i
361.1 −2.33136 + 1.34601i 0.500000 + 0.866025i 2.62349 4.54402i 1.04892i −2.33136 1.34601i −0.480608 0.277479i 8.74094i −0.500000 + 0.866025i −1.41185 2.44540i
361.2 −2.04113 + 1.17845i 0.500000 + 0.866025i 1.77748 3.07868i 3.69202i −2.04113 1.17845i 0.694498 + 0.400969i 3.66487i −0.500000 + 0.866025i 4.35086 + 7.53590i
361.3 −1.77441 + 1.02446i 0.500000 + 0.866025i 1.09903 1.90358i 3.35690i −1.77441 1.02446i 1.94594 + 1.12349i 0.405813i −0.500000 + 0.866025i −3.43900 5.95652i
361.4 1.77441 1.02446i 0.500000 + 0.866025i 1.09903 1.90358i 3.35690i 1.77441 + 1.02446i −1.94594 1.12349i 0.405813i −0.500000 + 0.866025i −3.43900 5.95652i
361.5 2.04113 1.17845i 0.500000 + 0.866025i 1.77748 3.07868i 3.69202i 2.04113 + 1.17845i −0.694498 0.400969i 3.66487i −0.500000 + 0.866025i 4.35086 + 7.53590i
361.6 2.33136 1.34601i 0.500000 + 0.866025i 2.62349 4.54402i 1.04892i 2.33136 + 1.34601i 0.480608 + 0.277479i 8.74094i −0.500000 + 0.866025i −1.41185 2.44540i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.i 12
13.b even 2 1 inner 507.2.j.i 12
13.c even 3 1 507.2.b.f 6
13.c even 3 1 inner 507.2.j.i 12
13.d odd 4 1 507.2.e.i 6
13.d odd 4 1 507.2.e.l 6
13.e even 6 1 507.2.b.f 6
13.e even 6 1 inner 507.2.j.i 12
13.f odd 12 1 507.2.a.i 3
13.f odd 12 1 507.2.a.l yes 3
13.f odd 12 1 507.2.e.i 6
13.f odd 12 1 507.2.e.l 6
39.h odd 6 1 1521.2.b.k 6
39.i odd 6 1 1521.2.b.k 6
39.k even 12 1 1521.2.a.n 3
39.k even 12 1 1521.2.a.s 3
52.l even 12 1 8112.2.a.cg 3
52.l even 12 1 8112.2.a.cp 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.i 3 13.f odd 12 1
507.2.a.l yes 3 13.f odd 12 1
507.2.b.f 6 13.c even 3 1
507.2.b.f 6 13.e even 6 1
507.2.e.i 6 13.d odd 4 1
507.2.e.i 6 13.f odd 12 1
507.2.e.l 6 13.d odd 4 1
507.2.e.l 6 13.f odd 12 1
507.2.j.i 12 1.a even 1 1 trivial
507.2.j.i 12 13.b even 2 1 inner
507.2.j.i 12 13.c even 3 1 inner
507.2.j.i 12 13.e even 6 1 inner
1521.2.a.n 3 39.k even 12 1
1521.2.a.s 3 39.k even 12 1
1521.2.b.k 6 39.h odd 6 1
1521.2.b.k 6 39.i odd 6 1
8112.2.a.cg 3 52.l even 12 1
8112.2.a.cp 3 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{12} - 17T_{2}^{10} + 195T_{2}^{8} - 1260T_{2}^{6} + 5963T_{2}^{4} - 15886T_{2}^{2} + 28561 \) Copy content Toggle raw display
\( T_{5}^{6} + 26T_{5}^{4} + 181T_{5}^{2} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 17 T^{10} + 195 T^{8} + \cdots + 28561 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} + 26 T^{4} + 181 T^{2} + 169)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{10} + 31 T^{8} - 28 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{12} - 41 T^{10} + 1207 T^{8} + \cdots + 2825761 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + T^{5} + 17 T^{4} - 42 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} - 21 T^{10} + 343 T^{8} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( (T^{6} + 49 T^{4} - 182 T^{3} + 2401 T^{2} + \cdots + 8281)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 2 T^{5} + 19 T^{4} - 28 T^{3} + \cdots + 841)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 174 T^{4} + 7985 T^{2} + \cdots + 38809)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} - 166 T^{10} + \cdots + 20200652641 \) Copy content Toggle raw display
$41$ \( T^{12} - 73 T^{10} + 4115 T^{8} + \cdots + 707281 \) Copy content Toggle raw display
$43$ \( (T^{6} + 15 T^{5} + 178 T^{4} + 623 T^{3} + \cdots + 1681)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 21 T^{4} + 98 T^{2} + 49)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 17 T^{2} + 66 T - 41)^{4} \) Copy content Toggle raw display
$59$ \( T^{12} - 68 T^{10} + \cdots + 116985856 \) Copy content Toggle raw display
$61$ \( (T^{6} - 13 T^{5} + 185 T^{4} + \cdots + 27889)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 213 T^{10} + 44155 T^{8} + \cdots + 2825761 \) Copy content Toggle raw display
$71$ \( T^{12} - 182 T^{10} + \cdots + 1698181681 \) Copy content Toggle raw display
$73$ \( (T^{6} + 306 T^{4} + 29301 T^{2} + \cdots + 851929)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} - 18 T + 27)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 62 T^{4} + 649 T^{2} + 1849)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} - 201 T^{10} + \cdots + 163047361 \) Copy content Toggle raw display
$97$ \( T^{12} - 587 T^{10} + \cdots + 7181161893361 \) Copy content Toggle raw display
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